4MA0 Higher January 2017 Paper 3R Q22
22 There are 12 sweets in a bag.
4 of the sweets are lemon flavour.
4 of the sweets are strawberry flavour.
4 of the sweets are orange flavour.
Luke takes at random 3 of the sweets.
Work out the probability that exactly 2 of the sweets that Luke takes are the same flavour.
(5)
| Scheme | Marks |
|---|---|
| \(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\left(= \dfrac{48}{1320} = \dfrac{2}{55}\right)\) oe | M1 |
| \(3 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\) or \(2 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\) | M1 |
| \(3 \times 2 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\) oe or \(3 \times 3 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\) | M1 |
| \(3 \times 3 \times 2 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{4}{10}\) oe | M1 |
| \(\dfrac{36}{55}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: or M2 for \(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{8}{10}\left(= \dfrac{96}{1320} = \dfrac{4}{55}\right)\) oe
M1: or M1 for \(3 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{8}{10}\) oe
M1: or M1 for \(3 \times 3 \times \dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{8}{10}\) oe
A1: oe eg. \(\dfrac{864}{1320}\) (0.65(45454...))
Alternative using 1 – (all different + all the same)
| Scheme | Marks |
|---|---|
| \(\dfrac{4}{12} \times \dfrac{4}{11} \times \dfrac{4}{10}\) or \(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{2}{10}\) | M1 |
| \(\dfrac{4}{12} \times \dfrac{4}{11} \times \dfrac{4}{10} \times 6\) or \(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{2}{10} \times 3\) | M1 |
| \(\dfrac{4}{12} \times \dfrac{4}{11} \times \dfrac{4}{10} \times 6\) and \(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{2}{10} \times 3\) | M1 |
| \(1 - \left[\left(\dfrac{4}{12} \times \dfrac{4}{11} \times \dfrac{4}{10} \times 6\right) + \left(\dfrac{4}{12} \times \dfrac{3}{11} \times \dfrac{2}{10} \times 3\right)\right]\) | M1 |
| \(\dfrac{36}{55}\) | A1 |
Notes
A1: oe eg. \(\dfrac{864}{1320}\) (0.65(45454...))
SC: With replacement (maximum marks M3)
| Scheme | Marks |
|---|---|
| \(3 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{4}{12}\left(= \dfrac{192}{1728} = \dfrac{1}{9}\right)\) or \(2 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{4}{12}\left(= \dfrac{128}{1728} = \dfrac{2}{27}\right)\) | M1 |
| \(3 \times 2 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{4}{12}\) oe or \(3 \times 3 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{4}{12}\) oe | M1 |
| \(3 \times 3 \times 2 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{4}{12}\) oe | M1 |
Notes
M1: or \(\dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{8}{12}\)
M1: or \(3 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{8}{12}\)
M1: M1 for \(3 \times 3 \times \dfrac{4}{12} \times \dfrac{4}{12} \times \dfrac{8}{12}\)